Oh, cross posted with your links, Christy.
I haven't time now unfortunately to discuss in detail , but both are interesting. I don't think either of those are of practical use for parents though.
The first study looked at exclusive breastfeeding to 9 months and found greater incidence of hypersensitivity. But there are 2 big problems with this.
Firstly: no-one is recommending exclusive breastfeeding to 9 months.
Secondly, this is an observational study and the big problem with these is reverse causation. In allergy research this happens because allergic families know they're allergic, and are much more likely to delay giving foods to babies. So, lots of studies show that babies who are weaned later are allergic, but you can't conclude anything because in many cases they were weaned later because they were allergic, not allergic because they were weaned later.
The other article isn't a research paper but an opinion paper - and it's always worth reading expert opinions - but all it really adds is to point out that the allergy/weaning issue is controversial.
There's a big trial going on that I'm aware of (likely more than one) which is properly randomised, and when the results come out it'll be very interesting to see what they show, because the reverse causation problem should be accounted for by randomising children to different treatment.
I'm just an interested reader, I should point out, so am not claiming any expertise - we're an allergic family and so this is of great interest to me. I find it really frustrating that there are no clear answers.